2.3 Stability estimates: linear case, Fourier-BDF2
2.3.4 Fourier-based BDF2-ADI stability: parabolic equation
tion (2.74). The result is
<(u, AB Du)≥ 1
2D|Bu|2− 1
8|D2u|2. (2.75) The lemma now follows by averaging equations (2.74) and (2.75).
tion (2.76).
We first note that a calculation similar to that leading to equation (2.55) shows that the Fourier-based BDF2-ADI scheme for (2.76) can be expressed in the form
Duˆ −∆t(α δxx+β δyyu+γ δxδy)u+ ∆t γ δxδyD2u+b(∆t)2αβ δxxδyyDu= 0. (2.79)
Letting
A=−∆t α δxx, B =−∆t β δyy, F =−∆t γ δxδy, L=A+B+F,
equation (2.79) becomes
Duˆ +Lu−F D2u+bAB Du= 0. (2.80)
Note that the operators A and B above do not coincide with the corresponding A and B operators in Section 2.3.3.
In view of the exactness relation (2.41) together with the Fourier differentiation operators (cf. (2.42)), it follows that A, B, AB and L are positive semidefinite oper- ators. Indeed, in view of equation (2.78), for example, we have
(u, L u) = − ∆t (2π)2
Z 2π 0
Z 2π 0
uN (α(uN)xx+β(uN)yy+γ(uN)xy) dx dy
≥0; (2.81)
similar relations for A, B and AB follow directly by integration by parts.
Finally we present yet another consequence of the parabolicity condition (2.77)
which will prove useful: for any grid function g we have
|F g|2 =γ2(∆t)2(δxδyg, δxδyg)≤4αβ(∆t)2(g, δx2δy2g) = 4(g, ABg). (2.82)
Thus, defining the seminorm
|u|P =p
(u, P u) (2.83)
for a given positive semidefinite operator P and using P =AB we obtain
|F g|2 ≤4|g|2AB. (2.84)
The following theorem can now be established.
Theorem 2.2: The solution u of the Fourier-based BDF2-ADI scheme (2.79) for equation (2.76) with initial conditions u0, u1 satisfies
1
4|un|2+1
4|eun+1|2+ 1
3|(Du)n|2AB +1 4
n
X
m=1
|D2u|2 +
n
X
m=1
|un|2L≤M
for n≥2, where
M =1
4|u1|2+ 1
4|eu2|2+ 1
3|u1|2AB + 3|u1|L− <(u1, F (Du)1) + 3|(Du)1|2+3
2 |(Du)1|2A+|(Du)1|2B
+1
3|(Du)1|2AB.
In particular, the scheme is unconditionally stable in the sense of equation (2.35).
Proof: Taking the inner product of (2.80) with u we obtain
0 = (u,Du) + (u, Lu)ˆ −(u, F D2u) +b(u, AB Du) (2.85)
= (I) + (II) + (III) + (IV),
where(I) = (u,Du),ˆ (II) = (u, Lu), etc. As in Theorem 2.1, we re-express the above
equation using telescoping and non-negative terms to obtain the desired energy bound.
The term (I) already occurs in the proof of Theorem 2.1; there we obtained the relation
<(I) = 1
4D(|u|2+|we|2) + 1
4|D2u|2, (2.86) where weis defined in (2.58). The term (II) =|u|2L, in turn, is non-negative by equa- tion (2.81) and needs no further treatment. The remaining two terms are considered in what follows.
(III): This term presents the most difficulty, sinceF is not positive semi-definite. In what follows the term (III) is re-expressed as a a sum of two quantities, the first one of which can be combined with a corresponding term arising from the quantity(IV) to produce a telescoping term, and the second of which will be addressed towards the end of the proof by utilizing Lemma 2.2.
Let v denote the time series obtained by shifting u backwards by one time step:
v ={vn =un−1 :n≥1}; (2.87)
clearly we have
Du=u−v and D2u=Du−Dv. (2.88)
Thus, using the finite difference product rule (2.51) and the second relation in (2.88) we obtain
(III) =−(u, F D(Du)) =−(u, D F(Du))
=−D(u, F Du) + (Du, F Du)−(Du, F D2u)
=−D(u, F Du) + (Du, F Dv).
Applying the Cauchy-Schwarz inequality and Young’s inequality (2.70) with r = 6
together with (2.84) we obtain
<(III)≥ −D<(u, F Du)− |Du| |F Dv|
≥ −D<(u, F Du)−3|Du|2− 1
12|F Dv|2
≥ −D<(u, F Du)−3|Du|2− 1
3|Dv|2AB. (2.89) The last term in the above inequality will be used to form the desired telescoping term with an associated expression in (IV) below.
(IV): Using the finite difference product rule (2.52) together with the fact that AB is a Hermitian positive semi-definite operator we obtain
<(IV) = 2
3<(u, AB Du) = 2
3<(Du, ABu)
= 1
3D(u, AB u) + 1
3(Du, AB Du)
= 1
3D|u|2AB +1
3|Du|2AB (2.90)
(see equation (2.83)). Substituting equations (2.86), (2.89) and (2.90) into equa- tion (2.85) and taking real parts, we obtain
0≥1
4D(|u|2+|we|2) + 1
4|D2u|2+|u|2L−D<(u, F Du)−3|Du|2 +1
3(|Du|2AB− |Dv|2AB) + 1
3D|u|2AB
=D 1
4|u|2+ 1
4|we|2 +1
3|u|2AB +1
3|Du|2AB− <(u, F Du)
+|u|2L+1
4|D2u|2−3|Du|2. (2.91)
Adding the time series (2.91) from m = 2 to n and using the identity wen =uen+1 we obtain
M1 ≥ 1
4|un|2+ 1
4|eun+1|2+ 1
3|un|2AB+1
3|(Du)n|2AB +
n
X
m=2
|un|2L
+1 4
n
X
m=2
|(D2u)n|2−3
n
X
m=2
|(Du)m|2− <(un, F (Du)n) (2.92)
where
M1 = 1
4|u1|2+1
4|eu2|2+1
3|u1|2AB+ 1
3|(Du)1|2AB − <(u1, F (Du)1).
Using Cauchy-Schwarz and Young’s inequalities along with the parabolicity rela- tion (2.84) and the fact thatF is a Hermitian operator, the last term−<(un, F (Du)n) in (2.92) is itself estimated as follows:
−<(un, F(Du)n) = −<(F un,(Du)n)
≥ −|F un||(Du)n|
≥ − 1
12|F un|2 −3|(Du)n|2
≥ −1
3|un|2AB−3|(Du)n|2.
Equation (2.92) may thus be re-expressed in the form 1
4|un|2 + 1
4|eun+1|2+1
3|(Du)n|2AB+
n
X
m=2
|un|2L+ 1 4
n
X
m=2
|D2u|2
≤ M1+ 3|(Du)n|2+ 3
n
X
m=2
|(Du)m|2. (2.93)
Finally, applying Lemma 2.2 below to the last two terms on the right-hand side of equation (2.93) we obtain
3|(Du)n|2+ 3
n
X
m=2
|(Du)m|2 ≤3M2,
where the constant M2 is given by equation (2.95). Using this inequality to bound the last two terms in equation (2.93) completes the proof of the theorem.
The following lemma, which provides a bound on sums of squares of the temporal difference Du, is used in the proof of Theorem 2.2 above.
Lemma 2.2: The solutionuof the Fourier-based BDF2-ADI scheme (2.79)for equa- tion (2.76) with initial conditions u0, u1 satisfies
|(Du)n|2+|un|2L+ 1
2 |(Du)n|2A+|(Du)n|2B
+
n
X
m=2
|(Du)m|2 ≤M2 (2.94)
for n≥2, where
M2 =|(Du)1|2+|u1|2L+ 1
2 |(Du)1|2A+|(Du)1|2B
. (2.95)
Proof: We start by taking the inner product of equation (2.80) with Du to obtain 0 = (Du,Du) + (Du, Lu)ˆ −(Du, F D2u) +b(Du, AB Du) (2.96)
= (I) + (II) + (III) + (IV).
We now estimate each of the terms (I) through (IV) in turn; as it will become apparent, the main challenge in this proof is to estimate the term(III).
(I): Using (2.48) and the finite difference product rule (2.52), (I)can be expressed in the form
<(I) =<(Du, Du+1 2D2u)
=|Du|2+ 1
4D|Du|2+ 1
4|D2u|2. (2.97) (II): Using equation (2.52) we obtain
<(II) = <(Du, Lu) = 1
2D(u, Lu) + 1
2(Du, L Du).
Since L=A+B+F we may write
<(II) = 1
2D|u|2L+ 1
2|Du|2A+B+1
2(Du, F Du). (2.98) The last term in this equation (which is a real number in view of the Hermitian character of the operatorF) will be used below to cancel a corresponding term in our estimate of (III).
(III): Using (2.87) together with the second equation in (2.88), (III) can be ex- pressed in the form
(III) = −(Du, F D2u)
=−1
2(Du, F Du) + 1
2(Du, F Dv)− 1
2(Du, F D2u). (2.99) As mentioned in the treatment of (II) above, the first term on the right-hand side of (2.99) will be used to cancel the last term in (2.98). Hence it suffices to obtain bounds for the second and third terms on the right-hand side of equation (2.99).
To estimate the second term in (2.99) we consider the relation 1
2(Du, F Dv) = 1
2γ∆t(Du, δxδyDv) =−γ
4∆t(δxDu, δyDv)−γ
4∆t(δyDu, δxDv), (2.100) which follows from the fact that δx and δy are skew-Hermitian operators. Taking real parts and applying the Cauchy-Schwarz and Young inequalities together with the parabolicity condition (2.77) we obtain
1
2<(Du, F Dv)≥ −
√αβ
2 ∆t 1 2
rα
β |δxDu|2+1 2
rβ
α|δyDv|2
!
−
√αβ
2 ∆t 1 2
rβ
α|δyDu|2+ 1 2
rα
β |δxDv|2
!
=− 1
4∆t(α|δxDu|2+β|δyDu|2)−1
4∆t(α|δxDv|2+β|δyDv|2)
=− 1
4|Du|2A+B−1
4|Dv|2A+B. (2.101)
To estimate third term in (2.99) we once again use the Cauchy-Schwarz and Young inequalities and we exploit the relation (2.84); we thus obtain
−1
2<(Du, F D2u) =−1
2<(F Du, D2u)
≥ −1
6|F Du|2− 3 8|D2u|2
≥ −2
3|Du|2AB− 3
8|D2u|2. (2.102) Taking the real part of (2.99) and using equations (2.101) and (2.102) we obtain the relation
<(III)≥ −1
2<(Du, F Du)− 1
4|Du|2A+B− 1
4|Dv|2A+B−2
3|Du|2AB− 3
8|D2u|2, (2.103) which, as shown below, can be combined with the estimates for (I), (II), and (IV) to produce an overall estimate that consists solely of non-negative and telescoping
terms—as desired.
(IV): In view of (2.83) we see that(IV)coincides with theP-seminorm ofDuwith P =AB,
<(IV) = (IV) = 2
3|Du|2AB, (2.104)
which, of course, is non-negative, and therefore this term does not require any further treatment.
To complete the proof of the lemma we take real parts in equation (2.96) and we substitute (2.97), (2.98), (2.103) and (2.104); the result is
0≥|Du|2+ 1
4D|Du|2− 1
8|D2u|2+1
2D|u|2L+ 1
4|Du|2A+B−1
4|Dv|2A+B
=|Du|2− 1
8|D2u|2+D 1
4|Du|2+ 1
2|u|2L+1
4|Du|2A+B
. (2.105)
The first two terms on the right-hand-side can be bounded by expanding |D2u|2 and using Cauchy-Schwarz and Young’s inequalities to obtain
|Du|2− 1
8|D2u|2 =|Du|2− 1
8|Du−Dv|2 (2.106)
=|Du|2− 1
8(|Du|2+|Dv|2) + 1
4<(Du, Dv) (2.107)
≥ |Du|2− 1
8(|Du|2+|Dv|2)−1
4|Du||Dv| (2.108)
≥ |Du|2− 1
4(|Du|2+|Dv|2) (2.109)
= 1
2|Du|2+1
4D|Du|2. (2.110)
Substituting this result into (2.105), we obtain
0≥1
2|Du|2 +D 1
2|Du|2+1
2|u|2L+ 1
4|Du|2A+B
, (2.111)
which, as needed, is expressed as a sum of non-negative and telescoping terms. Adding
the time-series (2.111) from m = 2 to n yields the desired equation (2.94), and the
proof is thus complete.
Remark 2.5: It is interesting to point out that Lemma 2.2 by itself implies a weak stability result that follows from equation (2.94) and the Cauchy-Schwarz inequality:
|un|=|u1+
n
X
m=2
(Du)m|
≤ |u1|+
n
X
m=2
|(Du)m|
≤ |u1|+ n
n
X
m=2
|(Du)m|2
!12
≤ |u1|+p
nM2, (2.112)
Theorem 2.2 provides a much tighter energy estimate than (2.112), of course.
2.3.4.1 Stability in non-periodic domain with Legendre collocation The stability result for the parabolic equation can easily be extended to a non-periodic setting using a Legendre polynomial collocation spatial approximation. Here we pro- vide the main necessary elements to produce the extensions of the proofs. Background on polynomial collocation methods may be found, e.g., in [57].
Under Legendre collocation we discretize the domainΩ = [−1,1]×[−1,1]by means of the N + 1 Legendre Gauss-Lobatto quadrature nodes xj = yj (j = 0, . . . , N) in each one of the coordinate directions, which defines the grid{(xj, yk) : 0≤j, k ≤N} (with x0 =y0 =−1and xN =yN = 1). For real-valued grid functions f = (fjk) and g = (gjk) we use the inner product
(f, g) =
N
X
j=0 N
X
k=0
wjwkfjkgjk, (2.113)
where w` (0 ≤ ` ≤ N) are the Legendre Gauss-Lobatto quadrature weights. The
interpolant fN of a grid functionf is a linear combination of the form
fN(x, y) =
N
X
j=0 N
X
k=0
fˆjkPj(x)Pk(y)
of Legendre polynomials Pj.
A certain exactness relation similar to the one we used in the Fourier case exists in the Legendre context as well. Namely, for grid functions f and g for which the product of the interpolants has polynomial degree ≤2N−1in the x(resp. y) variable, the j (resp. k) summation in the inner product (2.113) of the two grid functions is equal to the integral of their corresponding polynomial interpolants with respect to x (resp. y) [47, Sec. 5.2.1]—i.e.,
(f, g) =
N
X
k=0
Z 1
−1
fN(x, yk)gN(x, yk)dx, (2.114a) provided
deg fN(x, yk)gN(x, yk)
≤2N −1for all 0≤k ≤N, and
(f, g) =
N
X
j=0
Z 1
−1
fN(xj, y)gN(xj, y)dy, (2.114b) provided
deg fN(xj, y)gN(xj, y)
≤2N −1 for all 0≤j ≤N.
Thus, for example, defining the Legendre x-derivative operatorδx as the derivative of the Legendre interpolant (cf. (2.42)) (with similar definitions forδy,δxx,δyy etc.), the exactness relation (2.114a) holds whenever one or both of the grid functions f and g is a Legendre x-derivative of a certain grid function.
A stability proof for the parabolic equation with zero Dirichlet boundary condi-
tions
Ut=α Uxx+β Uyy+γ Uxy inΩ, U = 0 on∂Ω,
can now be obtained by reviewing and modifying slightly the strategy presented for the periodic case in Section 2.3.4. Indeed, the latter proof relies on the following properties of the spatial differentiation operators:
1. The discrete first and second derivative operators are skew-Hermitian and Her- mitian, respectively.
2. The operatorsA,B,LandABdefined in Section 2.3.4 are positive semi-definite.
Both of these results were established using the exactness relation between the dis- crete and integral inner products together with vanishing boundary terms arising from integration by parts—which also hold in the present case since the exactness relations (2.114) are only ever required to convert inner products involving deriva- tives, so that the degree of polynomial interpolants will satisfy the requirements of the relations (2.114). Since all other aspects of the proofs in Section 2.3.4 are inde- pendent of the particular spatial discretization or boundary conditions used, we have the following theorem:
Theorem 2.3: The stability estimate given in Theorem 2.2 also holds on the domain [−1,1] ×[−1,1] with homogeneous boundary conditions using the Legendre Gauss- Lobatto collocation method, where the inner products and norms are taken to be the Legendre versions instead.